Martin van Buren was born in the United States, but was he a natural-born citizen of the United States?

I was thinking about Van Buren yesterday because he's the answer to my favorite piece of presidential trivia—which president's first language wasn't English?1—and thus my contribution to the viral Twitter post about true pieces of history that sound fake. The other fun fact about Van Buren is that he was the first president born after the revolution—his birthday is December 5th, 1782—and he is often described as the "first president born a citizen of the United States." That's certainly correct in almost every sense—the first seven presidents were all clearly born as colonial British subjects2 and a great variety of legal documents either use the term "United States of America" or declare it to be the name of the new nation. The 7/4/1776 Declaration of Independence refers to the revolting colonies as the united States of America," albeit with united not capitalized. A 9/9/1776 resolution of the continental Congress) puts "United States" into formal practice to replace "United…